Quantum vortices in weakly coupled superfluids have a large healing length, so that many particles reside within the vortex core. They are characterized by topologically protected singular points, which in principal should keep their core structure rigid. I will describe how, in practice, the point singularity of a vortex deforms into a line singularity, in proportion with the Magnus force experienced by the vortex. The vortex structure is described by weak solutions of the Gross-Pitaevskii equation, similar to shock waves in hydrodynamics. I will discuss how the core deformation significantly affects many aspects of vortex dynamics. A striking example I will describe is the instability of the Abrikosov vortex lattice in the weak-coupling limit. All vortex singularities in the lattice spontaneously deform into finite cuts, which then order into superstructures.